One of the main operations performed for Algebraic Expression is Division. Deeply understand how division operations are performed on Algebraic Expressions and learn the easy way to solve Algebraic Expression Division Problems. Solving Division of Algebraic Expression Problems is the opposite process of Multiplication of Algebraic Expression.
Rule to Find Division of Algebraic Expression
If x is a variable and a, b are positive integers such that a > b then (x^a ÷ x^b) = x^(a − b).
Types of Algebraic Expression Division
There are different types of Division when it comes to Dividing Algebraic Expressions. They are as such
- Division of a Monomial by a Monomial
- Division of a Polynomial by a Monomial
- Division of a Polynomial by a Polynomial
How to Find Division of a Monomial by a Monomial?
1. The coefficients of the quotient of two monomials are equal to the quotient of their numerical coefficients which are multiplied by the quotient of their coefficients.
2. The variable part of the quotient of two monomials is equal to the quotient of the variables in the given monomials.
Rule:
Quotient of two monomials = (quotient of their numerical coefficients) x (the quotient of their variables)
Solved Examples
1. Divide 6x2y3 by -4xy
Solution:
Given that Divide 6x2y3 by -4xy
Use quotient law to solve the given problem.
xm ÷ xn = xm – n
x2/x = x^(2-1) = x
y3/y = y^(3-1) = y²
-(6/4) = -3/2
-3/2 x y²
The required expression is -3/2xy²
2. Divide 48x3yz2 by -8xyz
Solution:
Given that Divide 48x3yz2 by -8xyz
Use quotient law to solve the given problem.
xm ÷ xn = xm – n
x3/x = x^(3-1) = x²
y/y = 1
z2/z = z^(2-1) = z
(48/-8) = -6
-6x2z
The required expression is -6x2z
3. Divide -24x3yz3 by -6xyz2
Solution:
Given that Divide -24x3yz3 by -6xyz2
Use quotient law to solve the given problem.
xm ÷ xn = xm – n
x3/x = x^(3-1) = x²
y/y = y^(1-1) = y^0 = 1
z3/z2 = z^(3-2) = z
(-24/-6) = 4
4x2z
The required expression is 4x2z
How to Find Division of a Polynomial by a Monomial?
Division of a Polynomial by a Monomial can be calculated by dividing each term of the polynomial by the monomial.
1. Note down the polynomial and the monomial.
2. Consider polynomial as Dividend and monomial as the divisor.
3. Separate the terms of the polynomial (dividend) in the descending order of their exponents.
4. Finally, divide every term of the polynomial (dividend) by monomial.
5. Simplify and write the required expression.
Solved Examples
1. Divide 8x5 + 20x4 – 16x2 by 4x2
Solution:
Given that Divide 8x5 + 20x4 – 16x2 by 4x2
Here the dividend is 8x5 + 20x4 – 16x2 and the divisor is 4x2
(8x5 + 20x4 – 16x2) ÷ 4x2
Divide every term of the polynomial (dividend) by monomial.
(8x5 ÷ 4x2 ) +( 20x4 ÷ 4x2 ) – ( 16x2 ÷ 4x2
(8/4)(x^5/x^2) + (20/4)(x^4/x^2) – (16/4)(x^2/x^2)
2(x^5-2) + 5(x^4-2) – 4(x^2-2)
2x^3 + 5x^2 – 4
The required expression is 2x3 + 5x2 – 4
2. Divide 21x3y + 12x2y2 – 9xy by 3xy
Solution:
Given that Divide 21x3y + 12x2y2 – 9xy by 3xy
Here the dividend is 21x3y + 12x2y2 – 9xy and the divisor is 3xy
21x3y + 12x2y2 – 9xy ÷ 3xy
Divide every term of the polynomial (dividend) by monomial.
(21x3y ÷ 3xy) + (12x2y2 ÷ 3xy) – (9xy ÷ 3xy)
(21/3)(x3/x)(y/y) + (12/3)(x2/x)(y2/y) – (9/3)(xy/xy)
7(x^3-1)(y^1-1) + 4 (x^2-1)(y^2-1) – 3
7x^2 + 4xy – 3
The required expression is 7x2 + 4xy – 3
3. 10x + 20x2 + 30x4 – 50x5 ÷ by 5x
Solution:
Given that Divide 10x + 20x2 + 30x4 – 50x5 by 5x
Here the dividend is 10x + 20x2 + 30x4 – 50x5 and the divisor is 5x
Arrange the dividend in the descending order of their exponents.
( – 50x5 + 30x4 + 20x2 + 10x ) ÷ 5x
Divide every term of the polynomial (dividend) by monomial.
(-50/5)(x5/x) + (30/5)(x4/x) + (20/5)(x2/x) + (10/5)(x/x)
(-50/5)(x^5-1) + (30/5)(x^4-1) + (20/5)(x^2-1) + (10/5)(x^1-1)
-10x4 + 6x3 + 4x + 2
The required expression is -10x4 + 6x3 + 4x + 2
How to Find Division of a Polynomial by a Polynomial?
Get to know the Procedure on How to Divide Polynomial by a Polynomial in the below modules. They are along the lines
1. Note down the polynomial and the monomial.
2. Consider polynomial as Dividend and monomial as the divisor.
3. Separate the terms of the polynomial (dividend) in the descending order of their exponents.
4. Divide the first term of the polynomial (dividend) by the first term of the polynomial (divisor) and get the first term of the quotient.
5. Multiply every term of the divisor by the first term of the quotient and then subtract the result from the dividend.
6. In the next step, take the remainder if you have any and consider it as a new dividend then go with the above process.
7. Repeat the same process till you get the remainder as 0 or a polynomial of degree less than that of the divisor.
Solved Examples
1. Divide 18 – 12a² – 8a by (3 + 2a)
Solution:
Given that Divide 18 – 12a² – 8a by (3 + 2a)
Here the dividend is – 12a² – 8a + 18 and the divisor is 2a + 3
Multiply 6a with (2a + 3) then subtract the resultant expression from – 12a² – 8a + 18
-6a (2a + 3) = – 12a² – 18a
(– 12a² – 8a + 18) + (- 12a² – 18a) = 10a + 18
Now, multiply 5 with (2a + 3) then subtract the resultant expression from 10a + 18
5(2a + 3) = 10a + 15
10a + 18 – 10a – 15 = 3
Quotient = -6a + 5
Remainder = 3
Verification:
Dividend = divisor × quotient + remainder
– 12a² – 8a + 18 = (3 + 2a) × (-6a + 5) + 3
= -18a + 15 – 12a² + 10a + 3
= – 12a² – 8a + 18
– 12a² – 8a + 18 = – 12a² – 8a + 18
The required answer is -6a + 5
2. Divide 8x² + 12x + 4 by (4x + 4).
Solution:
Given that Divide 8x² + 12x + 4 by (4x + 4)
Here the dividend is 8x² + 12x + 4 and the divisor is 4x + 4.
Multiply 2x with (4x + 4) then subtract the resultant expression from 8x² + 12x + 4
2x(4x + 4) = 8x² + 8x
(8x² + 12x + 4) – (8x² + 8x) = 4x + 4
Multiply 1 with (4x + 4) then subtract the resultant expression from 4x + 4
1(4x + 4) = 4x + 4
(4x + 4) – (4x + 4) = 0
Quotient = 2x + 1
Remainder = 0
Verification:
Dividend = divisor × quotient + remainder
8x² + 12x + 4 = (4x + 4) × (2x + 1) + 0
= 8x² + 4x + 8x + 4 + 0
= 8x² + 12x + 4
8x² + 12x + 4 = 8x² + 12x + 4
The required answer is 2x + 1
3. Divide 3x² + 18x + 24 by (3x + 12)
Solution:
Given that Divide 3x² + 18x + 24 by (3x + 12)
Here the dividend is 3x² + 18x + 24 and the divisor is (3x + 12)
Multiply x with (3x + 12) then subtract the resultant expression from 3x² + 18x + 24
x(3x + 12) = 3x² + 12x
(3x² + 18x + 24) – (3x² + 12x) = 6x + 24
Multiply 2 with (3x + 12) then subtract the resultant expression from 6x + 24
2(3x + 12) = 6x + 24
(6x + 24) – (6x + 24) = 0
Quotient = x + 2
Remainder = 0
Verification:
Dividend = divisor × quotient + remainder
3x² + 18x + 24 = (3x + 12) × (x + 2) + 0
= 3x² + 6x + 12x + 24 + 0
= 3x² + 18x + 24
3x² + 18x + 24 = 3x² + 18x + 24
The required answer is x + 2
4. Divide 18x – 12x² + 2x³ – 4 by (2x – 4).
Solution:
Given that Divide 18x – 12x² + 2x³ – 4 by (2x – 4)
Here the dividend is 2x³ – 12x² + 18x – 4 and the divisor is (2x – 4)
Multiply x² with (2x – 4) then subtract the resultant expression from 2x³ – 12x² + 18x – 4
x² (2x – 4) = 2x³ – 4x²
(2x³ – 12x² + 18x – 4) – (2x³ – 4x²) = – 8x² + 18x – 4
Multiply -4x with (2x – 4) then subtract the resultant expression from – 8x² + 18x – 4
-4x (2x – 4) = -8x² + 16x
(- 8x² + 18x – 4) – (-8x² + 16x) = 2x – 4
Multiply 1 with (2x – 4) then subtract the resultant expression from 2x – 4
1 (2x – 4) = 2x – 4 – 2x + 4 = 0
Quotient = x² – 4x + 1
Remainder = 0
Verification:
Dividend = divisor × quotient + remainder
18x – 12x² + 2x³ – 4 = (2x – 4) × (x² – 4x + 1) + 0
= 2x³ – 8x² + 2x – 4x² + 16x – 4 + 0
= 2x³ – 12x² + 18x – 4
2x³ – 12x² + 18x – 4 = 2x³ – 12x² + 18x – 4
The required answer is x² – 4x + 1
5. Divide (87x – 18x² – 84) by (9x – 12).
Solution:
Given that Divide (87x – 18x² – 84) by (9x – 12)
Here the dividend is – 18x² + 87x – 84 and the divisor is (9x – 12)
Multiply -2x with (9x – 12) then subtract the resultant expression from – 18x² + 87x – 84
-2x (9x – 12) = – 18x² + 24x
(- 18x² + 87x – 84) – (- 18x² + 24x) = 63x – 84
Multiply 7 with (9x – 12) then subtract the resultant expression from 63x – 84
7 (9x – 12) = 63x – 84
(63x – 84) – (63x – 84) = 0
Quotient = -2x + 7
Remainder = 0
Verification:
Dividend = divisor × quotient + remainder
(87x – 18x² – 84) = (9x – 12) × (-2x + 7) + 0
= – 18x² + 63x + 24x – 84
= 87x – 18x² – 84
87x – 18x² – 84 = 87x – 18x² – 84
The required answer is -2x + 7
6. Divide (20x³- 16x² + 12x + 72) by (12 – 8x + 4x²)
Solution:
Given that Divide (20x³- 16x² + 12x + 72) by (12 – 8x + 4x²)
Here the dividend is (20x³- 16x² + 12x + 72) and the divisor is (12 – 8x + 4x²)
Multiply 5x with (4x² -8x + 12) then subtract the resultant expression from 20x³- 16x² + 12x + 72
5x (4x² -8x + 12) = 20x³- 40x² + 60x
(20x³- 16x² + 12x + 72) – (20x³- 40x² + 60x) = 24x² – 48x + 72
Multiply 6 with (4x² -8x + 12) then subtract the resultant expression from 24x² – 48x + 72
6 (4x² -8x + 12) = 24x² – 48x + 72
(24x² – 48x – 72) – (24x² – 48x + 72) = 0
Quotient = 5x + 6
Remainder = 0
Verification:
Dividend = divisor × quotient + remainder
(20x³- 16x² + 12x + 72) = (4x² -8x + 12) × (5x + 6) + 0
= 20x³ + 24x² – 40x² – 48x + 60x + 72
= (20x³- 16x² + 12x + 72)
(20x³- 16x² + 12x + 72) = (20x³- 16x² + 12x + 72)
The required answer is 5x + 6
7. Using division, show that (2x – 2) is a factor of (2x³ – 2).
Solution:
Given that Using division, show that (2x – 2) is a factor of (2x³ – 2)
Divide (2x³ – 2) by (2x – 2)
Here the dividend is (2x³ – 2) and the divisor is (2x – 2)
Multiply x with (2x – 2) then subtract the resultant expression from (2x³ – 2)
x (2x – 2) = 2x³- 2x
(2x³ – 2) – (2x³- 2x) = 2x – 2
Multiply 1 with (2x – 2) then subtract the resultant expression from (2x – 2)
(2x – 2) – (2x – 2) = 0
Quotient = x + 1
Remainder = 0
(2x – 2) divides (2x³ – 2). Therfore, (2x – 2) is the factor of (2x³ – 2)
8. Find the quotient and remainder when (14 + 30x – 26x² + 10x³) is divided by (8 – 6x + 2x²)?
Solution:
Given that Divide (14 + 30x – 26x² + 10x³) by (8 – 6x + 2x²)
Here the dividend is (14 + 30x – 26x² + 10x³) and the divisor is (8 – 6x + 2x²)
Rearrange the given expressions.
divide (10x³ – 26x² + 30x + 14) by (2x² – 6x + 8)
Multiply 5x with (2x² – 6x + 8) then subtract the resultant expression from (10x³ – 26x² + 30x + 14)
5x (2x² – 6x + 8) = 10x³ – 30x² + 40x
(10x³ – 26x² + 30x + 14) – (10x³ – 30x² + 40x) = 4x² – 10x + 14
Multiply 2 with (2x² – 6x + 8) then subtract the resultant expression from 4x² – 10x + 14
2 (2x² – 6x + 8) = 4x² – 12x + 16
(4x² – 10x + 14) – (4x² – 12x + 16) = 2x – 2
Quotient = 5x + 2
Remainder = 2x – 2
The Quotient is 5x + 2 and the Remainder is 2x – 2
9. Divide (30x⁴ + 51x³ – 186x² + 90x – 9) by (6x² + 21x – 3)
Solution:
Given that Divide (30x⁴ + 51x³ – 186x² + 90x – 9) by (6x² + 21x – 3)
Here the dividend is (30x⁴ + 51x³ – 186x² + 90x – 9) and the divisor is (6x² + 21x – 3)
Multiply 5x² with (6x² + 21x – 3) then subtract the resultant expression from (30x⁴ + 51x³ – 186x² + 90x – 9)
5x² (6x² + 21x – 3) = 30x⁴ + 105x³ – 15x²
(30x⁴ + 51x³ – 186x² + 90x – 9) – (30x⁴ + 105x³ – 15x²) = -54x³ – 171x² + 90x – 9
Multiply 9x with (6x² + 21x – 3) then subtract the resultant expression from -54x³ – 171x² + 90x – 9
-9x (6x² + 21x – 3) = -54x³ – 189x² + 27x
(-54x³ – 171x² + 90x – 9) – (-54x³ – 189x² + 27x) = 18x² + 63x – 9
Multiply 3 with (6x² + 21x – 3) then subtract the resultant expression from 18x² + 63x – 9
3 (6x² + 21x – 3) = 18x² + 63x – 9
(18x² + 63x – 9) – (18x² + 63x – 9) = 0
Quotient = 5x² – 9x + 3
Remainder = 0
Verification:
Dividend = divisor × quotient + remainder
(30x⁴ + 51x³ – 186x² + 90x – 9) = (6x² + 21x – 3) × (5x² – 9x + 3) + 0
= 30x⁴ -54x³ + 18x² -105x³ – 189x² +63x – 15x² +27x – 9
= (30x⁴ + 51x³ – 186x² + 90x – 9)
(30x⁴ + 51x³ – 186x² + 90x – 9) = (30x⁴ + 51x³ – 186x² + 90x – 9)
The required answer is 5x² – 9x + 3